Schanuel's conjecture (1960's) is a comprehensive transcendence statement about values of the exponential function. Ax-Schanuel is a functional analogue, due to Ax in 1971. Analogues of special cases of this result for Shimura varieties have been crucial ingredients in recent approaches to the Zilber-Pink Conjecture, which also has origins in Schanuel's Conjecture. I will describe these connections and recent work with Ngaiming Mok and Jacob Tsimerman giving an Ax-Schanuel theorem for Shimura varieties.